Determinacy of the selection games G_1(Σ,f) and G_{<ω}(Σ,f) in full generality

Determine whether the point-set selection games G_1(Σ,f) and G_{<ω}(Σ,f) are determined for every metric space X, every nonempty family Σ of nonempty subsets of X, and every real-valued function f: X → ℝ; that is, ascertain whether, without imposing additional structural assumptions on Σ or X, one of the two players always has a winning strategy in each of these games.

Background

The paper introduces two related selection games on a metric space X with a family Σ of nonempty subsets: G_1(Σ,f) and G_{<ω}(Σ,f). The authors prove these games are equivalent and show that under certain hypotheses (e.g., Σ dense and consisting of closed sets in a complete metric space), the games are determined, with determinacy linked to hereditary small oscillation and continuous restriction properties.

Beyond these specific settings, the authors explicitly state that it remains an open question whether the games are determined in full generality for arbitrary choices of Σ and f. They further pose a related existence question later in the paper about whether there is any example where G_1(Σ,f) is not determined.

References

We show that these two games are equivalent and, in several settings, determined. However, we leave an open question whether they are determined in general.

— Point-set games and functions with the hereditary small oscillation property  (2405.15263 - Balcerzak et al., 2024) in Section 1 (Introduction)